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The formula you have given here is the zero counting formula for all nontrivial zeros with positive imaginary part $\le T$ instead of the one that only counts zeros on the critical line.
I did find Lichtman's paper during the search while writing the answer. As I did not have communications with him before about such generalizations, I did not put his result in the answer.
@Wojowu Well, we already know that there are infinitely many even numbers that can be expressed as $p+q$ such that $p,q$ are prime and $|p-q|\le246$ due to Polymath.
Let $p_n$ be $n$'th prime. Then it is known that $p_{n+1}-p_n=O(p_n^\theta)$ for some $\theta<1$. The value of $\theta$ depends on subconvexity estimates of $\zeta\left(\frac12+it\right)$. More can be found in the wiki page
Have you tried playing with Jensen's formula and Borel-Caratheodory lemma? These are typically used to establish bounds for logarithmic derivatives of $\zeta(s)$ and $L(s,\chi)$.
